Discrete Laplacians on the hyperbolic space -- a compared study (2409.01211v2)

Abstract: The main motivation behind this paper stems from a notable gap in the existing literature: the absence of a discrete counterpart to the Laplace-Beltrami operator on Riemannian manifolds, which can be effectively used to solve PDEs. We consider that the natural approach to pioneer this field is first to explore one of the simplest non-trivial (i.e., non-Euclidean) scenarios, specifically focusing on the $2$-dimensional hyperbolic space $\mathbb{H}^2$. We present two variants of discrete finite-difference operator tailored to this constant negatively curved space, both serving as approximations to the (continuous) Laplace-Beltrami operator within the $\mathrm{L}^2$ framework. We prove that the discrete heat equation associated with both operators mentioned above exhibits stability and converges towards the continuous heat-Beltrami Cauchy problem on $\mathbb{H}^2$. Moreover, using techniques inspired from the sharp analysis of discrete functional inequalities, we prove that the solutions of the discrete heat equations corresponding to both variants of discrete Laplacian exhibit an exponential decay asymptotically equal to the one induced by the Poincar\'e inequality on $\mathbb{H}^2$. Eventually, we illustrate that a discrete Laplacian specifically designed for the geometry of the hyperbolic space yields a more precise approximation and offers advantages from both theoretical and computational perspectives. Furthermore, this discrete operator can be effectively generalized to the three-dimensional hyperbolic space.